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Simplifying 0.5n2 + -0.5n + -256 = 0 Reorder the terms: -256 + -0.5n + 0.5n2 = 0 Solving -256 + -0.5n + 0.5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -512 + -1n + n2 = 0 Move the constant term to the right: Add '512' to each side of the equation. -512 + -1n + 512 + n2 = 0 + 512 Reorder the terms: -512 + 512 + -1n + n2 = 0 + 512 Combine like terms: -512 + 512 = 0 0 + -1n + n2 = 0 + 512 -1n + n2 = 0 + 512 Combine like terms: 0 + 512 = 512 -1n + n2 = 512 The n term is -1n. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 512 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 512 + 0.25 Combine like terms: 512 + 0.25 = 512.25 0.25 + -1n + n2 = 512.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 512.25 Calculate the square root of the right side: 22.632940596 Break this problem into two subproblems by setting (n + -0.5) equal to 22.632940596 and -22.632940596.Subproblem 1
n + -0.5 = 22.632940596 Simplifying n + -0.5 = 22.632940596 Reorder the terms: -0.5 + n = 22.632940596 Solving -0.5 + n = 22.632940596 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 22.632940596 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 22.632940596 + 0.5 n = 22.632940596 + 0.5 Combine like terms: 22.632940596 + 0.5 = 23.132940596 n = 23.132940596 Simplifying n = 23.132940596Subproblem 2
n + -0.5 = -22.632940596 Simplifying n + -0.5 = -22.632940596 Reorder the terms: -0.5 + n = -22.632940596 Solving -0.5 + n = -22.632940596 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -22.632940596 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -22.632940596 + 0.5 n = -22.632940596 + 0.5 Combine like terms: -22.632940596 + 0.5 = -22.132940596 n = -22.132940596 Simplifying n = -22.132940596Solution
The solution to the problem is based on the solutions from the subproblems. n = {23.132940596, -22.132940596}
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